Question

1. which data set shows the least evidence of a linear relationship?
2. for which data set does the correlation coefficient r appear to be equal to 1?
3. for which data set is the correlation coefficient r closest to - 1?

Explanation:

Step1: Recall correlation - coefficient properties

The correlation coefficient \(r\) measures the strength and direction of a linear relationship. \(r = 1\) indicates a perfect positive - linear relationship, \(r=- 1\) indicates a perfect negative - linear relationship, and \(r\) close to \(0\) indicates a weak or no linear relationship.

Step2: Analyze for least linear relationship

A data - set with a weak or no linear relationship will have a correlation coefficient close to \(0\). Visually, the data points will be scattered randomly. Among the figures, if the data points are randomly scattered, it shows the least evidence of a linear relationship.

Step3: Analyze for \(r = 1\)

For \(r = 1\), the data points should lie exactly on a straight - line with a positive slope.

Step4: Analyze for \(r=-1\)

For \(r=-1\), the data points should lie exactly on a straight - line with a negative slope.

Answer:

1. Without seeing the actual figures, if the data points in a figure are randomly scattered, it shows the least evidence of a linear relationship.
2. If the data points in a figure lie exactly on a straight - line with a positive slope, the correlation coefficient \(r = 1\).
3. If the data points in a figure lie exactly on a straight - line with a negative slope, the correlation coefficient \(r=-1\).